A Marcinkiewicz–Zygmund type strong law of large numbers for non-negative random variables with multidimensional indices

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A Marcinkiewicz–Zygmund type strong law of large numbers for non-negative random

variables with multidimensional indices

Tibor Tómács

^*

Institute of Mathematics and Informatics Eszterházy Károly University, Eger, Hungary

tomacs.tibor@uni-eszterhazy.hu Submitted: September 2, 2019

Accepted: December 4, 2019 Published online: December 5, 2019

Abstract

In this paper a Marcinkiewicz–Zygmund type strong law of large numbers is proved for non-negative random variables with multidimensional indices, furthermore we give its an application for multi-index sequence of non- negative random variables with finite variances.

Keywords:Marcinkiewicz–Zygmund type strong law of large numbers, almost sure convergence, non-negative random variables, multidimensional indices MSC:60F15

1. Introduction

The Kolmogorov theorem and the Marcinkiewicz–Zygmund theorem are two fa- mous theorems on the strong law of large numbers for 𝑋𝑛 (𝑛 ∈ N) sequence of independent identically distributed random variables (see e.g.Loève[8]). By Kol- mogorov theorem, there exists a constant 𝑏 such that lim_𝑛→∞𝑆𝑛/𝑛 = 𝑏 almost surely if and only if E|𝑋1|<∞, where 𝑆𝑛 =∑︀𝑛

𝑘=1𝑋𝑘. If the latter condition is satisfied then 𝑏 = E𝑋1. By Marcinkiewicz–Zygmund theorem, if 0< 𝑟 <2 then

*The author’s research was supported by the grant EFOP-3.6.1-16-2016-00001 (“Complex im- provement of research capacities and services at Eszterhazy Karoly University”).

doi: 10.33039/ami.2019.12.001 http://ami.uni-eszterhazy.hu

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lim𝑛→∞(𝑆𝑛−𝑏𝑛)/𝑛^1/𝑟 = 0almost surely if and only if E|𝑋1|^𝑟<∞, where𝑏= 0 if0< 𝑟 <1, and𝑏= E𝑋1if1≤𝑟 <2.

Etemadi [1] proved that the Kolmogorov theorem holds for identically distributed and pairwise independent random variables, furthermore Kruglov [7]

extended the Marcinkiewicz–Zygmund theorem for pairwise independent case if 𝑟 <1.

Several papers are devoted to the study of the strong law of large numbers for multi-index sequence of random variables (see e.g.Gut[4],Klesov[5, 6],Fazekas [2], Fazekas, Tómács [3]). For example, Theorem 3.1 ofFazekas, Tómács [3]

extends Theorem 2 ofKruglov[7] for multi-index case.

In this paper the main result is Theorem 3.1, which is a Marcinkiewicz–Zygmund type strong law of large numbers for non-negative random variables with multidimensional indices. It is a generalization of Theorem 3.1 of Fazekas, Tómács[3]

in case n→ ∞. Furthermore we give an application (see Theorem 4.1) for multi- index sequence of non-negative random variables with finite variances. A special case of this result gives Theorem of Petrov[9].

2. Notation

Let N^𝑑 be the positive integer𝑑-dimensional lattice points, where 𝑑 is a positive integer. For n,m ∈ N^𝑑, n ≤ m is defined coordinate-wise, (n,m] = (𝑛1, 𝑚1]× (𝑛2, 𝑚2]× · · · ×(𝑛𝑑, 𝑚𝑑] is a𝑑-dimensional rectangle and|n|=𝑛1𝑛2· · ·𝑛𝑑, where n = (𝑛1, 𝑛2, . . . , 𝑛𝑑), m = (𝑚1, 𝑚2, . . . , 𝑚𝑑). ∑︀

n will denote the summation for alln∈N^𝑑. We also use1= (1,1, . . . ,1)∈N^𝑑 and 2= (2,2, . . . ,2)∈N^𝑑. Denote the integer part of𝑥real number by [𝑥].

We shall say thatlim_n→∞𝑎n= 0, where 𝑎n (n∈N^𝑑) are real numbers, if for all𝛿 >0 there existsN∈N^𝑑 such that|𝑎n|< 𝛿 ∀n≥N.

We shall assume that random variables𝑋n (n∈N^𝑑) are defined on the same probability space(Ω,ℱ,P). EandVarstand for the expectation and the variance.

Remark that a sum or a minimum over the empty set will be interpreted as zero (i.e.∑︀

n∈𝐻𝑎n= minn∈𝐻𝑎n= 0if𝐻 =∅).

3. The result

The following result is a generalization of Theorem 3.1 of Fazekas, Tómács [3]

in casen→ ∞.

Theorem 3.1. Let 𝑋n (n∈N^𝑑)be a sequence of non-negative random variables, let 𝑏n (n ∈ N^𝑑) be a sequence of non-negative numbers, 𝐵n = ∑︀

k≤n𝑏k, 𝑆n =

∑︀

k≤n𝑋k,𝑐 >0,𝐾∈N and0< 𝑟≤1. If

𝐵n−𝐵m≤𝑐(|n| − |m|) ∀n,m∈N^𝑑,n≥m,|n| − |m| ≥𝐾 (3.1)

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and ∑︁

n

1

|n|P(︁

|𝑆n−𝐵n|> 𝜀|n|^1/𝑟)︁

<∞ ∀𝜀 >0, (3.2)

then

n→∞lim

𝑆n−𝐵n

|n|^1/𝑟 = 0 almost surely. Proof. Let𝛿 >0,1< 𝛼 <(︀_𝛿

2𝑐 + 1)︀^1/3𝑑

and0< 𝜀 <^𝛿₂(︀_𝛿

2𝑐 + 1)︀−1/𝑟

, which imply

𝜀𝛼^3𝑑/𝑟+𝑐(𝛼^3𝑑−1)< 𝛿. (3.3)

Let𝑘𝑛 = [𝛼^𝑛] (𝑛∈N)andkn= (𝑘𝑛1, 𝑘𝑛2, . . . , 𝑘𝑛𝑑), wheren= (𝑛1, 𝑛2, . . . , 𝑛𝑑)∈ N^𝑑. It follows from the inequalities

∑︁

n

1

|n|P(︁

|𝑆n−𝐵n|> 𝜀|n|^1/𝑟)︁

≥∑︁

n

∑︁

h∈(kn,kn+1]

1

|h|P(︁

|𝑆h−𝐵h|> 𝜀|h|^1/𝑟)︁

≥∑︁

n

∑︁

h∈(k_n,k_n+1]

1

|kn+1| min

k∈(k_n,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

=∑︁

n

|kn+1−kn|

|kn+1| min

k∈(k_n,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

and condition (3.2) that

∑︁

n

|kn+1−kn|

|kn+1| min

k∈(k_n,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

<∞. (3.4) Since lim𝑛→∞(︀

1−𝛼^𝑛+1¹ −𝛼¹

)︀= 1−𝛼¹ >0, so (︀

1−𝛼^𝑛+1¹ −𝛼¹

)︀ > ^𝛼−1_2𝛼 except for finitely many𝑛∈N. This implies that there existsN0∈N^𝑑 such that

0<

(︂𝛼−1 2𝛼

)︂𝑑

<

∏︁𝑑

𝑖=1

(︂

1− 1 𝛼^𝑛^𝑖⁺¹ − 1

𝛼 )︂

=

∏︁𝑑

𝑖=1

𝛼^𝑛^𝑖⁺¹−1−𝛼^𝑛^𝑖 𝛼^𝑛^𝑖⁺¹

≤

∏︁𝑑

𝑖=1

[𝛼^𝑛^𝑖⁺¹]−[𝛼^𝑛^𝑖]

[𝛼^𝑛^𝑖⁺¹] = |kn+1−kn|

|kn+1| ∀n= (𝑛1, 𝑛2, . . . , 𝑛𝑑)≥N0. Hence

(︂𝛼−1 2𝛼

)︂𝑑 ∑︁

n≥N0

k∈(kmin_n,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

≤ ∑︁

n≥N0

|kn+1−kn|

|kn+1| min

k∈(kn,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

.

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By this inequality and (3.4), it follows that

∑︁

n≥N0

k∈(kminn,kn+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

<∞. (3.5)

Ifn≥N0 then there existsmn∈N^𝑑 such thatmn∈(kn,kn+1]and P(︁

|𝑆m_n−𝐵m_n|> 𝜀|mn|^1/𝑟)︁

= min

k∈(k_n,k_n+1]P(︁

|𝑆k−𝐵k|> 𝜀|k|^1/𝑟)︁

. Therefore, by (3.5) we have

∑︁

n≥N0

P(︁

|𝑆m_n−𝐵m_n|> 𝜀|mn|^1/𝑟)︁

<∞. (3.6)

By the Borel–Cantelli lemma, (3.6) implies that there exist N1 ∈ N^𝑑 and 𝐴∈ ℱ such thatN1≥N0,P(𝐴) = 1and

|𝑆mn(𝜔)−𝐵mn|

|mn|^1/𝑟 ≤𝜀 ∀n≥N1, ∀𝜔∈𝐴. (3.7) Henceforward let𝜔∈𝐴 be fixed.

Ifn≥N1andt∈(kn+1,kn+2], then byt∈(mn,mn+2], (3.7) and

|mn+2|^1/𝑟≥ |mn|^1/𝑟≥ |mn| we have

𝑆t(𝜔)−𝐵t

|t|^1/𝑟 ≥𝑆m_n(𝜔)−𝐵m_n+2

|mn+2|^1/𝑟

=𝑆m_n(𝜔)−𝐵m_n

|mn|^1/𝑟

|mn+2|^1/𝑟 −𝐵mn+2−𝐵mn

|mn+2|^1/𝑟

≥ −𝜀−𝐵m_n+2−𝐵m_n

|mn| . (3.8)

Ifn= (𝑛1, 𝑛2, . . . , 𝑛𝑑)≥N0 andmn= (m⁽¹⁾n ,m⁽²⁾n , . . . ,m^(𝑑)n )then [𝛼^𝑛^𝑖]<m^(𝑖)_n ≤[𝛼^𝑛^𝑖⁺¹].

On the other handm^(𝑖)n ∈N, hence we get

𝛼^𝑛^𝑖 <m^(𝑖)_n ≤𝛼^𝑛^𝑖⁺¹. (3.9) This inequality implies

|mn+2| − |mn|>

∏︁𝑑

𝑖=1

𝛼^𝑛^𝑖⁺²−

∏︁𝑑

𝑖=1

𝛼^𝑛^𝑖⁺¹

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= (𝛼^𝑑−1)

∏︁𝑑

𝑖=1

𝛼^𝑛^𝑖⁺¹

>(𝛼^𝑑−1)𝛼^𝑛¹ ∀n= (𝑛1, 𝑛2, . . . , 𝑛𝑑)≥N0.

Sincelim𝑛→∞𝛼^𝑛 =∞, therefore𝛼^𝑛≥𝐾(𝛼^𝑑−1)⁻¹except for finitely many values of𝑛∈N. Hence there existsN2∈N^𝑑 such that N2≥N1 and

|mn+2| − |mn|>(𝛼^𝑑−1) 𝐾

𝛼^𝑑−1 =𝐾 ∀n≥N2. This inequality implies by (3.1), that

𝐵m_n+2−𝐵m_n ≤𝑐(|mn+2| − |mn|) ∀n≥N2. (3.10) Using (3.9) we have

|mn+2|

|mn| ≤

∏︁𝑑

𝑖=1

𝛼^𝑛^𝑖⁺³

𝛼^𝑛^𝑖 =𝛼^3𝑑 ∀n= (𝑛1, 𝑛2, . . . , 𝑛𝑑)≥N2. (3.11) Hence (3.8), (3.10), (3.11) and (3.3) imply, that if n≥N2 and t∈(kn+1,kn+2], then

𝑆t(𝜔)−𝐵t

|t|^1/𝑟 ≥ −𝜀−𝐵mn+2−𝐵mn

|mn| ≥ −𝜀−𝑐

(︂|mn+2|

|mn| −1 )︂

≥ −𝜀−𝑐(𝛼^3𝑑−1)≥ −𝜀𝛼^3𝑑/𝑟−𝑐(𝛼^3𝑑−1)>−𝛿. (3.12) If n≥N2 and t∈(kn+1,kn+2], then byt∈(mn,mn+2],|mn|^1/𝑟 ≥ |mn|, (3.7), (3.11), (3.10) and (3.3), we have

𝑆t(𝜔)−𝐵t

|t|^1/𝑟 ≤𝑆mn+2(𝜔)−𝐵mn

|mn|^1/𝑟

=𝑆m_n+2(𝜔)−𝐵m_n+2

|mn+2|^1/𝑟

|mn|^1/𝑟 +𝐵m_n+2−𝐵m_n

|mn|^1/𝑟

≤𝑆m_n+2(𝜔)−𝐵m_n+2

|mn+2|^1/𝑟

|mn|^1/𝑟 +𝐵m_n+2−𝐵m_n

|mn|

≤𝜀𝛼^3𝑑/𝑟+𝑐

(︂|mn+2|

|mn| −1 )︂

≤𝜀𝛼^3𝑑/𝑟+𝑐(𝛼^3𝑑−1)< 𝛿.

This inequality and (3.12) imply

|𝑆t(𝜔)−𝐵t|

|t|^1/𝑟 < 𝛿 ∀n≥N2,t∈(kn+1,kn+2]. (3.13) If t ≥kN2+1+1, then there exists n≥ N2 such that t ∈ (kn+1,kn+2]. Hence (3.13) implies

|𝑆t(𝜔)−𝐵t|

|t|^1/𝑟 < 𝛿 ∀t≥kN2+1+1.

Therefore the statement is proved.

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4. An application for multi-index sequence of non- negative random variables with finite variances

In this section we give an application of Theorem 3.1. In case𝑑=𝑟= 1, this result gives Theorem ofPetrov[9].

Theorem 4.1. Let 𝑋n (n∈N^𝑑) be a sequence of non-negative random variables with finite variances, 𝑆n=∑︀

k≤n𝑋k,𝑐 >0,𝐾∈N and0< 𝑟≤1. If

E𝑆n−E𝑆m≤𝑐(|n| − |m|) ∀n,m∈N^𝑑,n≥m,|n| − |m| ≥𝐾 (4.1)

and ∑︁

n

Var𝑆n

|n|^1+2/𝑟 <∞, (4.2)

then

n→∞lim

𝑆n−E𝑆n

|n|^1/𝑟 = 0 almost surely.

Proof. With notation 𝑏k = E𝑋k and 𝐵n =∑︀

k≤n𝑏k = E𝑆n, (4.1) implies (3.1).

On the other hand, if𝜀 >0, then the Chebyshev inequality and (4.2) imply

∑︁

n

1

|n|P(︁

|𝑆n−𝐵n|> 𝜀|n|^1/𝑟)︁

≤∑︁

n

1

|n|

Var_|n|^𝑆1/𝑟ⁿ

𝜀² =𝜀⁻²∑︁

n

Var𝑆n

|n|^1+2/𝑟 <∞. Therefore (3.2) holds. Hence, using Theorem 3.1, we have that the statement is true.

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